Who Is Obligated to Wear Tefillin?
There is an obligation for men over the age of Bar Mitzvah to wear Tefillin daily.  The obligation of Tefillin is mentioned four times in the Torah.  The details of the laws are described below:
Who is obligated and who exempt from wearing Tefillin?
- All men above the age of 13 are required to put on Tefillin daily. 
- The common Minhag is that a boy under the age of 13 wears Tefillin only 2 or 3 months before his Bar Mitzvah.
- Tefillin is a Mitzvah Aseh SheHaZman Grama (time bound mitzvah) and so women are exempt. Even though usually women are allowed to fulfill mitzvot from which they are exempt, it is preferable for women not to fulfill this mitzvah of Tefillin.  Some maintain strongly that women should not wear Tefillin. 
- One who is shirtless cannot wear Tefillin. 
- One who is unable to control his promiscuous thought is exempt from Tefillin. 
- Rambam Sefer HaMitzvot (Aseh 12 and 13).
- The Torah mentions Tefillin twice when recalling the exodus from Egypt as in Exodus 13:9 it reads "And it shall be for a sign for you upon your hand, and for a memorial between your eyes, that the law of God may be in your mouth; for with a strong hand did God bring you out of Egypt." and Exodus 13:16 "And it shall be for a sign upon your hand, and as totafot between your eyes; for with a mighty hand did God bring us forth out of Egypt." and twice in the first two paragraphs of Kriyat Shema as in Deuteronomy 6:8 it says "And you shall bind them as a sign upon your arm, and they shall be as totafot between your eyes." and lastly in Deuteronomy 11:18 "You shall put these words of mine on your heart and on your soul; and you shall tie them for a sign upon your arm, and they shall be as totafot between your eyes."
- The Rambam (Sefer HaMitzvot Mitzvah Aseh 12 and 13) and Sefer HaChinuch (421) record this as a mitzvah that applies to every male individual even nowadays. Rambam there says based on the gemara in Menachot 44a that one who is obligated to wear Tefillin but does not, transgresses eight commandments (4 for the hand and 4 for the head).
- Kitzur Shulchan Aruch 10:24. See Yechave Daat 2:4 for a variety of Minhagim
- S”A and Rama 38:3, Mitzvat Nashim (pg 35). Gra in his comments to 38:3 writes that it is forbidden for women to wear Tefillin. See Targum Yonatan Ben Uziel on Devarim 22:5 who includes a woman wearing Tefillin in the category of Lo Yehey Kli Gever Al Isha.
In a letter dated Adar I 5774 (February 2014), Rav Hershel Schachter explains his position on this topic of women wearing Tefillin as follows:
- The Rabbis of the Mishna felt the need to display their disagreement with the Sadducees even when the Tzeduki opinion was a stringency, such as it was regarding Parah Adumah. Nonetheless, because the Rabbis noticed that their opinion was anti-Torah SheBaal Peh, in cases of disagreement the Rabbis made their position very evident.
- When developing new Minhagim such as women wearing Tefillin, Tzitzit, or Kippah, there is an obvious need to consult with a leading posek of the generation. It is a mistake to think that everyone who went to Yeshiva and knows how to find the right seforim, sometimes by searching online, should be able to decide halacha. Real questions of minhag and halacha should be presented to a major posek, who deals with the difficult halachic questions of the generation.
- Women wearing Tefillin is a symbol of the Conservative movement and it is crucial that the Orthodox community distinguishes itself. It is irrelevant to what extent woman can volunteer by the mitzvah of Tefillin because in our context, this is a clear symbol of the Conservative movement. As a precedent, Rav Soloveitchik absolutely forbad performing a Bat Mitzvah in the middle of davening similar to a Bar Mitzvah, the way the Conservatives did.
- Additionally, see Rabbi Mansour on dailyhalacha.com on this topic who also writes clearly that women should not wear Tefillin. See also Kaf Hachaim 38:9
- See also Rabbi Moshe Feinstein (Sh"t Iggerot Moshe OC 4:49) where he opposes any attempt to allow women to wear Tefillin.
- Kitzur Shulchan Aruch of Rav Rephael Baruch Toledano siman 10, who is obligated in tefillin and who is exempt, seif 2
- Kitzur Shulchan Aruch of Rav Rephael Baruch Toledano, siman 10, who is obligated in tefillin and who is exempt, seif 5